Numerical Stability of Path Tracing in Polyhedral Homotopy Continuation Methods

Authors:
S. Kim;M. Kojima
Affiliations:
Ewha Women’s University, Department of Mathematics, 11-1 Dahyun-dong, Sudaemoon-gu, 120-750, Seoul, Korea;Tokyo Institute of Technology, Department of Mathematical and Computing Sciences, 2-12-1 Oh-Okayama Meguro-ku, 152-8552, Tokyo, Japan
Venue:
Computing
Year:
2004

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Abstract

The reliability of polyhedral homotopy continuation methods for solving a polynomial system becomes increasingly important as the dimension of the polynomial system increases. High powers of the homotopy continuation parameter t and ill-conditioned Jacobian matrices encountered in tracing of homotopy paths affect the numerical stability. We present modified homotopy functions with a new homotopy continuation parameter s and various scaling strategies to enhance the numerical stability. Advantages of employing the new homotopy parameter s are discussed. Numerical results are included to illustrate the improved performance of the presented techniques.